Problem: Let $p(x) = 2x - 7$ and $q(x) = 3x - b$.  If $p(q(4)) = 7$, what is $b$?
Solution: Since $q(4) = 3\cdot 4 - b = 12-b$, we can write $p(q(4)) = 7$ as $p(12-b) = 7$.  Since $p(x) = 2x-7$, we have $p(12-b) = 2(12-b) - 7 = 17 - 2b$. Substituting this into $p(12-b) = 7$ gives $17-2b =7$, from which we have $b = \boxed{5}$.